Infinite Dust: the Counterintuitive Cantor Set

Chana Lyubich

Often, we study smooth, differentiable functions with the assumption that they are an adequate mathematical tool to describe our world. However, they are not always sufficient. A well-known example of this is the coastline paradox, which observes that the length of a coastline is not well-defined: rather, it depends on how the measurement was made. This is because a coastline has fractal properties, and does not resemble a differentiable curve. Indeed, fractals can be used to describe many concrete phenomena in nature, despite often having strange, counterintuitive properties!

One of the first examples of a fractal is the Cantor Set, which was discovered by Georg Cantor in 1883. (Incidentally, Georg Cantor is also the founder of set theory.) While the construction of the Cantor Set is deceptively simple, it results in some seemingly incompatible properties.

We can construct the Cantor Set $K$ as follows:

Let us take the segment $[0, 1]$ and remove the open middle third (that is, we remove the interval $(\frac{1}{3}, \frac{2}{3})$ ). We will label this first gap as $G^0$ and the two remaining segments as $I_{0}^{1}$ and $I_{1}^{1}$ . In this labeling method, the superscript indicates the “level” of the segment and the subscript indicates whether it is the left (0) or right (1) segment. Next, remove the open middle third interval from each of the remaining segments. We will label these two new gaps as $G_{0}^1$ and $G_1^1$ . The remaining intervals will be denoted $I_{00}^{2}$ , $I_{01}^{2}$ , $I_{10}^{2}$ , $I_{11}^{2}$ from left to right. In this way, we can construct the Cantor Set by repeatedly removing the open middle third of each remaining interval. This can be visualized with the following sketch of the first few levels and labels:

Thus, if we denote $I^n$ as the union of all remaining segments on level $n$ , then the Cantor Set is defined as:

$\displaystyle K = \bigcap_{n = 1}^{\infty} I^n$ .

In other words, it is the set of the remaining points after removing the open middle third interval from each level.

While the construction seems quite straightforward, the Cantor Set has some remarkably counterintuitive properties. In particular, despite having zero length, the Cantor Set has the same cardinality as the real numbers!

The fact that the Cantor Set has zero length should not be too surprising, considering that it is constructed by repeatedly removing intervals from $[0, 1]$ . Since we know that the length of the original segment is 1, the length of the Cantor set is given by 1 – (total length of all the gaps).

We know that the length of the first gap $G_0$ is $\frac{1}{3}$ .
Then, the length of $G_0^1 + G_1^1$ is $\frac{1}{9}+\frac{1}{9} = \frac{2}{9}$ .
The next four gaps each have length $\frac{1}{27}$ , so their total length is $\frac{4}{27}$ .

We see that adding the lengths of all the gaps gives us the following geometric series:

total gap length = $\frac{1}{3} + \frac{2}{3^2} + \frac{2^2}{3^3} + \cdots$

The sum of this geometric series is 1. Therefore, the total length of all the gaps is 1, and hence, the length of the Cantor Set must be zero.

And yet, despite its zero length, the Cantor Set is uncountable! To see this, let us take some point $x \in K$ . Then, we can say that $x$ is given by the intersection of a specific nest of intervals, denoted:

$\{x\} = \bigcap_{n=1}^{\infty}I^{n}_{i_{1}i_{2} \dots i_{n}}$ .

Recall, in this labeling system, each $i_n$ is either 0 or 1 and each time we descended to a new level, we added a new subscript to indicate whether we went left or right. The intersection of these intervals contains exactly one point (in this case, $x$ ). Therefore, we can map each point $x \in K$ to its unique sequence of labeling indices $i_1i_2i_3 \dots i_n i_{n+1}\dots$ . For example, the point $x = 0$ corresponds to the sequence $0000\dots$ However, the set of infinite sequences of 1s and 0s is uncountable! (This can be shown with Cantor’s proof by diagonal method). Therefore, the points in the Cantor Set must also be uncountable!

This is incredible when we notice that the number of gaps in the Cantor Set is, in fact, not uncountable. (This is because the gaps are mutually disjoint and non-degenerate.) Therefore, the set of all gaps is countable, and hence, the set of all endpoints of gaps must also be countable. This means that the vast majority of points in the Cantor Set are not the endpoints of the gaps! This is an extraordinary result, as intuitively, it would seem that the Cantor Set ought to consist only of the endpoints. And yet, the endpoints make up only a (relatively speaking) tiny subset of the Cantor Set as a whole!

It’s interesting to note that historically, physicists were quite resistant to the notion that the Cantor Set could be relevant to science. Yet, in the 20th century, Benoit Mandelbrot demonstrated that Cantor dusts can be used to model concrete systems, such as excess noise in transmission channels in various electrical engineering problems.

Thus, in addition to being beautiful in construction, the Cantor Set illustrates how fractals can play an essential role in our understanding of the natural world we live in. I hope this brief glimpse into the world of fractals displays their elegance and “uncountable” power!

Sources

Mandelbrot, Benoit B. The Fractal Geometry of Nature. 1982.

One Comment


May 30, 2023
djl9295

Really well done, Chana! I have actually always wondered about how approximate measurements are made for natural objects such as a coastline or leaves from a tree. Fractals are a part of many things in nature and Cantor’s Set is both an interesting and necessary method of thinking about them. It was difficult for me at first to wrap my head around the idea of the Cantor set having length = 0 but still being uncountable. However, I also ended up using the idea of Cantor’s diagonal method in my blog post as well. It was much easier for me to understand the uncountable argument when discussing the infinite sequences of 1s and 0s. If I am understanding this correctly, there are an uncountable number of points in the Cantor set but a countable number of endpoints? Is this related to the idea that every interval on the real numbers R is uncountable even if we are given the endpoints? This is a very interesting extension into countable vs. uncountable sets and how cardinality can be a very complicated topic. Thanks for sharing!

Sources

Chana

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